GOALS AND STRATEGIES FOR ESTIMATING TRENDS IN LANDBIRD
ABUNDANCE
JONATHAN BART,1 USGS Forest and Rangeland Ecosystem Science Center, Snake River Field Station, 970 Lusk Street, Boise,
ID 83706, USA
KENNETH P. BURNHAM, Colorado Cooperative Fish and Wildlife Research Unit, Room 201, Wagar Building, Colorado State University, Fort Collins, CO 80523, USA
ERICA H. DUNN, Canadian Wildlife Service, National Wildlife Research Centre, 100 Gamelin Boulevard, Hull, PQ K1A 0H3,
Canada
CHARLES M. FRANCIS, Canadian Wildlife Service, National Wildlife Research Centre, 100 Gamelin Boulevard, Hull, PQ K1A
0H3, Canada
C. JOHN RALPH, Redwood Sciences Lab, U.S. Forest Service, 1700 Bayview Drive, Arcata, CA 95521, USA
Abstract: Reliable estimates of trends in population size are critical to effective management of landbirds. We propose
a standard for considering that landbird populations are adequately monitored: 80% power to detect a 50% decline
occurring within 20 years, using a 2-tailed test and a significance level of 0.10, and incorporating effects of potential
bias. Our standard also requires that at least two-thirds of the target region be covered by the monitoring program.
We recommend that the standard be achieved for species’ entire ranges or for any area one-third the size of the
temperate portions of Canada and the United States, whichever is smaller. We applied our approach to North American Breeding Bird Survey (BBS) data. At present, potential annual bias for the BBS is estimated at ±0.008. Further,
the BBS achieves the monitoring standard for only about 42% of landbirds for which the BBS is considered the most
effective monitoring approach. Achieving the proposed monitoring target for ≥80% of these species would require
increasing the number of BBS—or similar survey—routes by several-fold, a goal that probably is impractical. We
suggest several methods for reducing potential bias and argue that if our methods are implemented, potential bias
would fall to ±0.003. The required number of BBS or similar routes would then be 5,106, about 40% more than in the
current BBS program. Most of the needed increases are in 15 states or provinces. Developing a comprehensive landbird monitoring program will require increased support for coordination of the BBS (currently 2 people) and new
programs for species that are poorly covered at present. Our results provide a quantitative goal for long-term landbird monitoring and identify the sample sizes needed, within each state and province, to achieve the monitoring goal
for most of the roughly 300 landbird species that are well suited to monitoring with the BBS and similar surveys.
JOURNAL OF WILDLIFE MANAGEMENT 68(3):611–626
Key words: birds, monitoring, North American Breeding Bird Survey, population trends, surveys, trend estimation.
Population size monitoring has provided critical information for nearly all major—and thousands of minor—wildlife issues during the past
several decades. Examples in which knowledge of
trends in population size have been critical to the
success of management programs include (1)
identification of pesticides as a serious threat to
wildlife (Carson 1962); (2) recovery of species
from pesticide impacts (Sheail 1985); (3) declines
of individual species such as spotted owls (Strix
occidentalis; Gutiérrez et al. 1995) and sage grouse
(Centrocercus urophasianus; Schroeder et al. 1999);
(4) declines of groups of species such as those in
eastern thickets, grasslands, and western riparian
habitats (Askins 2000); and (5) the recovery of
species under management such as the peregrine
falcon (Falco pereginus; White et al. 2002), Kirtland’s warblers (Dendroica kirtlandii; Mayfield
1992), and many waterfowl (Williams et al. 2002).
1
E-mail: jbart@eagle.boisestate.edu
Despite the success of many past monitoring
programs, much opportunity exists for making
programs even more useful (Downes et al. 2000,
Williams et al. 2002). We propose a quantitative
goal for landbird monitoring programs and
describe strategies for achieving the goal for
North American landbirds. We believe that a
quantitative goal is needed to design a comprehensive program, identify needed resources, and
measure progress. Our work is based on Butcher
et al. (1993), who proposed a monitoring goal,
and on recent work by Partners in Flight
(Downes et al. 2000, Pashley et al. 2000).
Substantial literature (summarized by Williams
et al. 2002) discusses the importance of relating
monitoring methods to management goals. For
example, quantitative models may be used to
assess costs and probabilities of different management errors and to identify optimal survey
methods and sample sizes in accordance with the
results. We are concerned with multispecies landbird surveys, and the integration of monitoring
611
612
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
and management on such a broad scale is necessarily incomplete. We have identified broad management goals, such as detecting declines, and we
believe that surveys can and should be designed
to help achieve these goals. No specific management programs can be envisaged, however,
because the surveys cover hundreds of species
and management actions will not—and cannot—
be designed until declines, and their causes, have
been identified.
We addressed 3 questions: (1) What should the
accuracy target be in programs to estimate trends
in landbird population size? (2) Which landbird
species and populations in North America warrant coverage by monitoring programs, and
which of these species are best monitored by the
BBS (Sauer et al. 2001) and similar programs?
(The phrase “and similar programs” means multispecies landbird surveys, using point counts conducted at about the same time that the BBS is
conducted, which can be used to supplement the
BBS). (3) How many of the populations best
monitored by the BBS are adequately covered
now, and how many routes would be needed to
provide adequate coverage for most of these populations? We also briefly discuss other measures
needed to develop a comprehensive program for
landbird monitoring.
METHODS
Expression for Power
Butcher et al. (1993) suggested that a reasonable accuracy target for trend-monitoring programs is 80% power to detect a 50% decline
occurring within 20 years. We evaluated this suggestion by deriving an expression for power, estimating its components, and exploring alternatives to the recommendation by Butcher et al.
(1993). Power usually is calculated under the
assumption that bias is zero. With trend estimates, however, this is often not a reasonable
assumption (Williams et al. 2002). One approach
for incorporating potential bias in the analysis is
to establish lower and upper limits (i.e., bl and bu)
for potential bias and to use these in deciding
whether an observed trend provides a reliable
indication of the direction of change in the population. For example, if bias in the estimated
annual rate of change is assumed to be as much
as ±1%, then bl = –0.01 and bu = 0.01. If the estimated annual rate of change in population size is
<1, we only conclude that the population has
declined if the observed decline is significantly
J. Wildl. Manage. 68(3):2004
less than 1 + bl (0.99 in the example). If the trend
estimate is >1, we only conclude that the population has increased if the observed increase is significantly greater than 1 + bu (1.01 in the example). Incorporation of potential bias into the
analysis in this manner affects power to detect a
change. We derived an expression for power that
incorporates limits for potential bias (Appendix A).
We present the expression in terms of the
required sample size (n; e.g., BBS routes), with
simple random selection of routes, to ensure that
power is at least 1 – β. The equation is
,
(1)
where sd is the sample standard deviation of within-route trends, α is the level of significance, d is
the duration of the survey in years, and C is the
change in population size during the d years
(e.g., for a 40% decline, C = –0.4).
Three Major Questions
Question (1): What should the accuracy target be in
programs to estimate trends in landbird population
size?—We addressed this question using BBS data.
We estimated sd in Expression (1) for d = 5, 10, 15,
and 20 years, for each of the 133 species recorded
on >500 BBS routes during 1982–2001. For d < 20,
we estimated sd for all sequential series of d years
and then calculated the average of these values
for each species. Periods that are too short may
yield misleading estimates of the long-term trend
due to cycles in the counts about the long-term
trend. We investigated this issue by calculating the
85, 95, and 99% confidence intervals (CI) for each
trend. We used a version of the trend estimation
method described in Bart et al. (2003) suitable for
stratified sampling and recorded the frequency
with which these CIs included the trend estimated
from the 20-year data set. We restricted the investigation to 2-tailed procedures on the assumption
that detecting increases, as well as decreases,
often is important. Reasonable values for C were
identified by determining the frequency distribution of 20-year trends for the 133 selected species.
We established values for bl and bu by reviewing
studies that identified sources of potential bias
and estimating the magnitude of these biases. We
then assessed the degree to which different
sources would be additive or complementary.
In many long-term monitoring programs, large,
continuous portions of the area of interest lay
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
outside the sampling frame (i.e., the set of locations that might be surveyed). For example, the
range of many North American species extends
into the boreal region, where few BBS routes are
conducted. We refer to this situation as incomplete “coverage.” We were unable to identify a
completely realistic model to explore the likely
magnitude of bias with incomplete coverage, but
we derived an approximate expression (Appendix A). We analyzed BBS data from the 6 U.S.
Fish and Wildlife Service (USFWS) regions that
together cover the coterminous United States.
We assumed that abundance for a given species
was equal across regions, and that all or none of
each region was covered by the surveys. In this
case, the probability that bias exceeds any given
level G, is approximately
,
(2)
where Z is a standard normal variable, n1 is the
number of USFWS regions covered by the survey,
and σ2 is the variance of the true trends among
the 6 regions. We derived an expression for σ2
that separated and removed sampling error
(Appendix A) and estimated its value for each of
the 133 selected species. We then plotted bias as
a function of how much of the range was covered,
and we used the results to develop a guideline for
how much of the study region needs to be covered by the surveys to avoid substantial bias.
Question (2): Which landbird species and populations in North America warrant coverage by monitoring
programs, and which species are best monitored by the
BBS and similar programs?—We identified landbirds that warrant monitoring using the general
principle that we should monitor species we would
try to conserve if we knew they were declining or
changing in other undesirable ways. According to
this principle, we would ideally monitor all landbird species that occur regularly in North America,
but we would not attempt to monitor species that
only occur rarely in North America. An initial list
was prepared that included all landbird species
with range maps in a popular field guide (National
Geographic Society 1999). A committee of landbird specialists, appointed by the Partners in Flight
Monitoring Working Group, then revised the list
in accordance with the general principle above.
Many of these species have substantial breeding
populations in both temperate and northern
(boreal and arctic) regions. Different survey methods must be used in boreal areas. We therefore dis-
613
tinguished separate temperate and northern populations for species that breed in both areas. Based
on results from our analysis of Question (1), we
defined separate temperate and northern populations if the temperate and northern regions
each contained >33% of the species’ range.
Many species that do breed in temperate regions
are not well suited to monitoring with the BBS. We
identified these species by defining 6 landbird survey methods and identifying species that will be
best monitored using the BBS and similar—rather
than other—methods. This analysis produced a
list of species that warrant monitoring and an
identified subset of these species that will be best
monitored with the BBS and similar programs.
Question (3): How many of the populations best monitored by the BBS are adequately covered now, and how
many routes would be needed to provide adequate coverage for most of these populations?—The proposed
monitoring target was expressed as the standard
error of the estimated trend. Trend estimates are
often needed for smaller areas than the entire
range, especially for species with wide ranges. We
considered different sized areas and concluded
that meeting the accuracy target for regions onethird the size of the temperate regions of Canada
and the United States represented a reasonable
trade-off between the difficulty of obtaining estimates for small areas and the need for estimates
within parts of the range. We obtained rangewide
estimated standard errors for population trends
for each landbird from Sauer et al. (2001). Next,
we converted these to the standard errors that
would have been achieved with a survey area onethird the size of the temperate region of Canada
and the United States (unless the species range
was smaller than this area). Finally, we recorded
whether the resulting standard error was less
than or greater than the threshold standard error
required to achieve the proposed monitoring
objective. This process produced a list of landbird species that currently are adequately monitored by the BBS. We also reported the number
of rangewide estimates that met the proposed
accuracy target to provide an indication of the
extra cost of meeting the accuracy target at the
smaller spatial scale.
Using Expression (1), we estimated how many
additional BBS or similar routes would be needed to achieve the monitoring objective for most
landbirds best monitored by the BBS. The term
sd in Expression (1) varies among species, and we
did not have sufficient data to estimate this quan-
614
Table 1. Frequency distribution for standard deviation (sd)
based on 133 species well surveyed by the North American
Breeding Bird Survey.a
sd
<0.060
0.059–0.080
0.081–0.140
0.141–0.200
>0.200
a
No. of
species
Proportion
Cumulative
proportion
2
19
85
18
9
0.01
0.14
0.65
0.13
0.07
0.01
0.15
0.80
0.93
1.00
Data collected during 1980–1999 were used to estimate sd.
tity for most species. We therefore calculated values of sd for a sample of well-surveyed species and
used the eightieth quantile as the value for sd.
This gave us a conservative value for sd. The other
terms in Expression (1) were specified in the proposed monitoring goal and did not vary among
species. We were thus able to calculate a single,
estimated sample size (no. of BBS or similar
routes) needed to achieve the accuracy target.
For the i th species, the density (yi ) of routes
needed to achieve the monitoring objective may
be expressed as
,
(3)
where n is the needed number of routes (not
species specific); Ai is the area covered by the
range of species i, or the portion of the range for
which the trend estimate is obtained; and pi is the
proportion of routes, within the area Ai, on which
the ith species is recorded at least 4 times (the
minimum for estimating trends recommended
by Bart et al. 2003). We selected 14 species with
small to medium-sized ranges, estimated the size
of each species’ breeding range using maps provided by Project WILDSPACE (Welsh et al. 1999),
and determined pi by analyzing BBS data using
the trend estimation method of Bart et al. (2003).
We then used Expression (3) to estimate the
needed density of routes for each of these species
with Ai equal to the species’ entire range or oneTable 2. Frequency distribution of trends during 1980–1999 for
133 landbird species well surveyed by the North American
Breeding Bird Survey.
Changea
(%)
No. of
species
Proportion
Cumulative
proportion
<25
26–50
51–75
>75
77
37
14
5
0.58
0.28
0.11
0.04
0.58
0.86
0.96
1.00
a
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Estimated increase or decrease during 1980–1999.
third the temperate region of Canada and the
United States (approx 3,000,000 km2), whichever
was smaller. Achieving the accuracy target for all
species using only the BBS and similar programs
would be difficult and probably is not a wise use
of resources. We selected a threshold value (y) as
the yi such that the monitoring objective would
be achieved for 80% of the species best monitored using the BBS and similar programs. This
density of routes was multiplied by the area of
each province and state to obtain the target, minimum number of BBS or similar routes. We
summed province- and state-specific values to
obtain the needed total number of routes.
RESULTS AND DISCUSSION
Question (11): What should the accuracy target be
in programs to estimate trend in landbird population size?
With d = 20 years, 80% of the values of sd were
≤0.14 and 93% were ≤0.20 (Table 1). Approximately 86% of the values of C were ≤50%, and
96% of the values were ≤75% (Table 2). In choosing the specifications for power, examining the
trade-off between probabilities of Type I and
Type II errors may be useful. Type I errors occur
when a true null hypothesis is rejected (“false
alarm” errors); Type II errors occur when a false
null hypothesis is not rejected (declines are
missed). Managers can protect themselves against
1 type of error by increasing the risk of the other
error. For example, suppose the conditions
described in Table 3 applied (d = 20, sd = 0.14, bl
= bu = 0, n = 100), and that managers were equally concerned with a false alarm and with not
detecting a 50% decline. Setting the significance
level at 0.15 would result in these 2 errors being
about equally likely. If a false alarm was viewed as
more serious than missing a decline, then the sigTable 3. Probabilities of Type I (C = 0) and Type II (C > 0) errors
in relation to level of significance for detecting population
declines. Standard deviations (sd ) estimated from data on
landbirds collected in the North American Breeding Bird Survey from 1980 to 1999.a
Significance
level
0%
0.05
0.10
0.15
0.20
0.05
0.10
0.15
0.20
a
From n =
(
20-yr decline (C ) in survey results
25%
50%
75%
0.83
0.73
0.66
0.60
sd (Zα/2 + Zβ)
| (C + 1)1/d – 1 | – (bu – bl )
0.14, bl = bu = 0, and n = 100.
0.32
0.22
0.16
0.13
2
)
0.00
0.00
0.00
0.00
with d = 20, sd =
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Table 4. Probabilities of Type I (C = 0) and Type II (C > 0) errors
in relation to number of routes (n) for detecting declines in
landbird species. Standard errors (se) estimated using North
American Breeding Bird Survey data from 1980 to 1999.a
No. of
routes
0%
100
200
300
400
0.10
0.10
0.10
0.10
a
From n =
(
20-yr decline (C ) in survey results
25%
50%
0.82
0.74
0.66
0.59
0.48
0.22
0.10
0.04
2
sd (Zα/2 + Zβ)
|(C + 1)1/d – 1 | – (bu – bl )
)
Table 6. Samples sizes required for 80% power to detect a
50% population decline in relation to standard deviation (sd ) of
the trend; sd estimated using North American Breeding Bird
Survey data form 1980 to 1999.a
75%
sd
0.05
0.04
0.00
0.00
0.00
0.08
0.14
0.20
0.30
43
133
271
609
with d = 20, sd =
615
a
From n =
(
Significance level (α)
0.10
0.15
34
104
213
480
29
88
179
404
2
sd (Zα/2 + Zβ)
|(C + 1)1/d – 1 | – (bu – bl )
)
0.20
25
76
156
350
with d = 20 and bl
0.20, bl = bu = 0, and α = 0.10.
= bu = 0. Note that n increases as sd 2.
nificance level might be set at 0.10 or 0.05. Note,
however, that with the 0.05 level, the chance of
missing a 50% decline is approximately 1 in 3 so
the survey would not be very effective in identifying declining species.
Sample size has a major effect on the relative
probabilities of Type I and Type II errors (Table 4).
As the number of routes increases from 100 to
400, the probability of missing a 50% decline
declines from 48 to 4%.
The value of sd also has a substantial influence
on the probability of Type I and Type II errors.
For example, under the conditions described in
Table 5 (d = 20, n = 300, bl = bu = 0, α = 0.05), the
probability of missing a 50% decline increases
from 1% with sd = 0.08 to 80% with sd = 0.30. Sample sizes required to achieve 80% power to detect
a 50% decline increase rapidly with increasing sd
(Table 6). Values of sd as low as 0.08 were rare in
the BBS dataset, so structuring a goal for power
around this level would not be useful. Similarly,
attempting to achieve high power for all species
(sd = 0.30) would be extremely difficult.
The change to be detected also has a major
influence on required samples sizes (Table 7).
For example, under the conditions in Table 7 (d =
20, sd = 0.20, bl = bu = 0, β = 0.20), nearly 6 times
the sample size is required to detect a 25%
change as to detect a 50% change.
The length of time during which a given change
(e.g., 50%) occurs also affects required sample
sizes. In Expression (1), for a given value of C, as
d becomes smaller, the term |(C + 1)1/d – 1| increases, which tends to reduce the required sample size. For example, with C = –0.50, |(C + 1)1/d
– 1| = 0.034 with d = 20 but is 0.045 with d = 15, an
increase of 32%. However, less data are collected
with a shorter duration, and this tends to increase
the sd and offset the reduction in |(C + 1)1/d – 1|.
The average increase in the sd for a 15-year survey, compared to a 20-year survey, was 16% and
for a 10-year survey was 33% (Table 8). The net
effect of these 2 changes is that for fixed values of
C, the required sample size declines as duration
decreases. For example, with sd = 0.20, significance = 0.10, and C = –0.50, the sample size
required for power of 80% is 213 for d = 20 and
166 for d = 15, a decline of 22%. This tendency
Table 5. Probabilities of Type I (C = 0) and Type II (C > 0) errors
in relation to the standard deviation (sd) of the trends as estimated for landbird species using North American Breeding
Bird Survey (BBS) data from 1980 to 1999.a
Proportion of species
with smaller
20-yr decline (C ) in survey results
sd
values of sd b
0%
25%
50%
75%
0.08
0.14
0.20
0.30
a
From n =
0.15
0.05
0.05
0.99
(
0.05
0.05
0.05
0.05
0.57
0.83
0.89
0.93
sd (Zα/2 + Zβ)
| (C + 1)1/d – 1 | – (bu – bl )
0.01
0.32
0.60
0.80
2
)
0.00
0.00
0.08
0.39
Table 7. Sample sizes required for 80% power to detect population declines of 25–75% in relation to level of significance.
Standard deviations (sd) estimated using data on landbird species in the North American Breeding Bird Survey from 1980 to
1999.a
0.05
0.10
0.15
0.20
with d = 20, n =
300, bl = bu = 0, and α = 0.05.
b Based on 133 species considered to be well studied by
BBS.
20-yr decline (C ) in survey results
25%
50%
75%
Significance
level
a
From n =
1,540
1,212
1,022
885
(
270
213
180
155
sd (Zα/2 + Zβ)
| (C + 1)1/d – 1 | – (bu – bl )
0.20, bl = bu = 0, and β = 0.20.
70
55
46
40
2
)
with d = 20, sd =
616
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Table 8. Relative standard deviation (sd ) and confidence interval (CI) coverage as a function of survey duration (d ; in years).
Analysis based on data for landbird species in the North American Breeding Bird Survey data from 1980 to 1999.
d
Relative sd
(sdd /sd20 )
5
10
15
2.15
1.33
1.16
Proportion of CIs that included
the 20-yr trend
0.85 CI
0.95 CI
0.99 CI
0.48
0.52
0.71
0.63
0.66
0.81
0.74
0.77
0.90
also means that a smaller change can be detected
with a given sample size when the survey period
is shorter. For example, with sd = 0.20, significance = 0.10, d = 20, and n = 213, the power to
detect a 50% decline would be 80%. With d = 15,
and assuming sd was 0.234 (a 16% increase), power
would be 80% to detect a decline of 45.5%. Thus,
for a given sample size, level of significance, and
power, a slightly smaller decline can be detected if
the decline occurs within 15 years rather than within 20 years. Also, a given decline will be detected
with higher power if it occurs in fewer years.
Reducing the duration of a survey increases the
risk that the estimated trend during the survey period will be a poor estimator of the 20-year trend due
to cycles. This risk rises sharply as duration decreases (Table 8). For example, only 71% of the calculated 85% CIs (based on 15 years), included the 20year trend, and only 52% of the calculated 85% CIs
(based on 10 years), included the 20-year trend.
Thus, trends based on 5 or 10 years were poor predictors of the 20-year trend, and even 15-year trends
often were quite different from 20-year trends.
Guidelines.—Selecting a reasonable target for
power depends on making several choices. We
suggest the following guidelines:
(1) The duration of the survey should be 20
years, in part because trends during shorter periods often are poor estimators of the longer trend
due to cycles (Table 8). Further, the gain achieved
by using a shorter period is small (e.g., 80%
power to detect a decline of 45.5% rather than
50% in the previous example), and large (e.g.,
50%) declines seldom occur in <20 years.
(2) We should assume the sd will be about 0.14.
Approximately 80% of the species had smaller values of sd (Table 6). Thus, a sample size that achieves
the accuracy target with sd = 0.14 will achieve the
target for most of the species. Achieving the target
for ≥90% of the species, while desirable, approximately doubles the required sample size.
(3) The goal of the survey should be to detect a
decline of about 50%. Detecting smaller declines
J. Wildl. Manage. 68(3):2004
(e.g., 25%) with power high enough to be useful
would be extremely expensive (Table 6), and
detecting these declines could only be justified if
explicit management objectives required that
level of precision and could justify the costs. If
the goal of detecting a 50% decline with 80%
probability is achieved, then we will have even
higher power to detect larger declines.
Potential Bias.—We have placed much emphasis in
this analysis on acknowledging bias. Bias usually is
difficult to estimate rigorously and often has been
ignored in the past. We believe, however, that analyses of avian trend data should include an explicit
discussion of bias and that an upper limit for its
effects on accuracy should be established. Our
rationale is that any use of trend (or any other)
estimates requires an assumption about bias.
A recent review of the BBS (O’Connor et al.
2000) identified 3 major potential sources of bias:
differences between regionwide and roadside
population trends, changes in observer detection
rates, and bias due to analytic methods. We discuss each of these sources of bias.
Two studies have compared regionwide and
roadside trends, both using change in habitat as
a surrogate for avian population change. Bart et
al. (1995) studied change in proportion of an
Ohio, USA, landscape covered by forest and
found little difference between the roadside and
regionwide trends. Further, Bart et al. (1995)
found no significant differences between the
annual rates of change in forest cover regionwide
and within 280 m of roads, suggesting that (in
this study area) bias due to restricting surveys to
roadsides was probably was <0.005.
Keller and Scallan (1999) studied regionwide
and roadside changes between 1963 and 1988 in
6 habitats and 12 habitat features (e.g., single
family homes) in Ohio and Maryland, USA. In
Ohio, none of the trends in habitats differed significantly (with α = 0.05) between the on-road
and off-road study areas. In Maryland, urban
habitat increased significantly faster along roads
(annual rate = 0.069) than off roads (0.039), and
agricultural habitats decreased faster along roads
(0.014 vs. 0.004). Of the habitat features measured, houses, buildings, and associated features
(e.g., driveways) increased significantly faster
along roads in both Ohio and Maryland than off
roads, but no other differences were significant.
The lack of significant results for most habitats
and habitat features suggests that bias would be
small except for birds dependent on human
developments. Thus, 2 studies that estimated the
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TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
bias in trend estimates due to surveys being
restricted to roadsides both found annual bias
probably was <0.005 for most species. While more
studies of this sort are needed and cases with larger bias can undoubtedly be found, absolute bias
from this source presently appears to be <0.005.
Change in the skill of BBS observers was studied by Sauer et al. (1994) and James et al. (1996).
In both studies, the investigators concluded that
a trend has occurred in average observer detection rates. This finding led to the development of
methods that remove certain trends in detection
rates (Link and Sauer 1994). Link and Sauer
(1994) noted that their model assumes that
observer-specific detection rates do not change
through time, whereas evidence exists for both a
learning effect (numbers detected in the first
year tend to be lower than numbers detected in
subsequent years [Kendall et al. 1996]) and for
senescence (numbers decline as observers’ hearing and perhaps vision decline). Both problems
are potentially serious. For example, if the population was stable, detection rates were 20% lower
the first year, and observers collected data for 7
years, then the estimated annual rate of change
would be 1.02 and bias would be 0.02, a figure
high enough to cause serious errors in estimates.
Bias caused by observer senescence would equal
the long-term trend in the detection rate for all
observers, which also could be substantial if
senescence is affecting a large fraction of the
observers. Both sources of bias could be reduced
by training and evaluation programs and perhaps
by developming models that incorporate estimates of their effects. Thus, while these problems
potentially are serious at present, they probably
can be reduced to low levels (e.g., absolute bias
<0.003) with the implementation of improved
analytic methods.
Bias due to analytic methods can be divided
into 2 sources: statistical bias that exists even if all
assumptions of the model are met, and bias due to
using a model whose assumptions are not fully met
by the data. Link and Sauer (1994) studied statistical bias in a route regression approach. Bias increased as trends diverged from 1.0 and with
decreasing abundance. Bias exceeded 0.005 only
for positive trends and for species recorded on average less than once per BBS route. Statistical bias
thus appears to be quite small for this approach.
The effects of model failure are harder to
assess. Link and Sauer (1994) pointed out that
their model assumed constant within-observer
detection rates, but this assumption may be false.
617
They also identified nonlinear trends within
routes and weighting factors as possible additional sources of bias. These factors deserve additional study (in part to increase precision), but they
should not produce substantial bias. Bart et al.
(2003) studied bias in a “linear method” for trend
estimation and in the route regression approach
of Link and Sauer (1994) by using real datasets to
establish actual trends and estimating these trends
by sampling with replacement (which simulates a
large population having the same actual trend as
the sample). Bart et al. (2003) studied 2 datasets,
1 from the BBS and 1 from the International
Shorebird Survey (Brown et al. 2001). With the
BBS data, bias was negligible (<0.002) with both
methods (Bart et al. 2003). With the shorebird
dataset, bias was negligible with the linear method
but was substantial (exceeding 0.01 for some species) with the route regression approach, probably because of large variation in numbers recorded within sites in the shorebird dataset (Bart et al.
2003). The general conclusion from these studies
is that analytic methods with negligible bias due
to the estimation methods probably can be found
for most datasets.
Another source of bias in the trend estimate
that probably has been of some significance in
the past—and may become much more significant in the future—is change in phenology. Hundreds of studies during the past decade have indicated that phenologies of many species are
changing (reviewed in Parmesan and Yohe 2003).
In general, species are starting their breeding
season earlier, though sometimes the reverse is
true. Changes in phenology change the frequency distribution of detection rates since most birds
have very different detection rates at different
stages in their nesting period. While this effect
could cause a substantial bias if global changes
continue to accelerate, adjusting the timing of
counts or estimating the change due to phenology should be possible based on intensive surveys
at a subset of locations (e.g., participating national wildlife refuges). A related source of bias is
change in survey dates (regardless of change in
phenology). Little investigation has addressed
whether the distribution of dates has been stable
throughout the history of the BBS.
In summary, more work is needed to establish
upper limits for bias in trend estimates based on
the BBS and other surveys. At present, however,
the few studies that have been completed suggest
that the bias probably will be small in most cases
if careful attention is given to discovering and
618
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Table 9. Influence of unacknowledged bias on the probability of
a Type I error (rejecting the null hypothesis when it is true).a
|Bias|
Assumed
0.000
0.005
0.010
0.05
0.05
0.05
Probability
Actual
Assumed
0.050
0.079
0.170
0.10
0.10
0.10
Actual
(bu – bl)
25%
Decline
50%
75%
0.100
0.142
0.264
0.000
0.004
0.008
0.016
0.024
0.53
0.31
0.14
0.01
0.00
1.00
0.99
0.96
0.73
0.30
1.00
1.00
1.00
1.00
1.00
a From
B
P (Type I error | bl, bu undefined ) = P Z < – Xα/ 2 – se(r )
(
)
B
+ P Z < – Xα/ 2 – se(r ) , with sd = 0.20 and n = 400.
(
Table 10. Influence of acknowledging bias on power to detect
a declinea in landbird species.
)
reducing sources of potential bias. We can reasonably assume that restricting the surveys to
roadsides, unacknowledged change in observer
skill, and bias due to the analytic method might
each contribute bias of up to 0.005. However, the
direction of the bias should not be the same in all
cases, so these biases should cancel out each
other to some extent. This suggests that a reasonable estimate for the upper limit of annual
bias in BBS estimates of trend at present is
±0.008. This estimate is admittedly crude, and
bias will obviously differ between species, but we
feel that making an estimate and using it when
establishing the monitoring goal is better than
ignoring the issue, which amounts to assuming
that bias is zero—a value even less defensible
than ±0.008. We therefore suggest using ±0.008 as
the limits for bias in BBS estimates until better
information leads to new, perhaps species-specific, values. If the bias-reduction methods we identified are implemented, then we can reasonably
assume that absolute bias could be reduced to
<0.003, and perhaps to even lower levels.
The presence of bias also affects tests, confidence intervals, and power. Unacknowledged
bias has a major effect on tests when the null
hypothesis is true (i.e., the population is stable)
over the study period. Bias tends to make rejecting the null hypothesis much more likely than
the nominal significance level. For example, if
the significance level is set at 0.05 but bias is
±0.01, then the actual probability of committing a
Type I error is 0.17, more than 3 times the nominal rate (Table 9). Power to detect a decline
depends on whether the bias is negative or positive. Negative bias makes rejecting the null
hypothesis more likely because the effects of the
decline are exaggerated. With positive bias, however, power is lessened because the bias tends to
mask the decline. These examples illustrate why
we recommend establishing limits for bias and
a Probability of obtaining a significant estimated trend that is
<1. From
|R – 1 | – | B – bp |
Power with bl , bu defined = P Z > Zα/ 2 –
,
se(r)
with d = 20, sd = 0.14, n = 400, and α = 0.05.
(
)
including the potential bias in tests, and therefore in the power calculations.
Acknowledging potential bias reduces power. For
example, under the conditions in Table 10, power
to detect a 50% decline falls from essentially 100%
when potential bias is 0.0 to only 73% when bu – bl
= 0.016, as occurs if bl = –0.008 and bu = 0.008, the
limits we suggested for BBS data. If bu – bl = 0.024,
power is only 30% to detect a 50% decline.
Combinations of the variables in Expression (1)
that would result in 80% power to detect a 50%
decline during 20 years are shown in Table 11.
Reducing potential bias from ±0.008 to ±0.003
reduces required sample sizes by about 58%.
Thus, if this reduction could be achieved by allocating some resources to the bias reduction,
higher power would be achieved as long as sample size was not reduced more than 58%. Some of
the bias reduction methods involve only the
analysis (e.g., eliminating first-year effects and
using habitat-based models to reduce roadside
bias), and thus should have low costs. Other
methods, such as double sampling to estimate
detection rates, will reduce sample size (assuming resources are fixed), and their feasibility is
unknown at present. Nonetheless, reducing bias
to ±0.003 seems at least possible.
Table 11. Sample size required to achieve 80% power to detect
a 50% population decline occurring within 20 years, in relation
to standard deviation (sd ) of the trends, potential bias, and significance level. Values used for sd were identified by analyzing
landbird data in the North American Breeding Bird Survey from
1980 to 1999.
Standard
deviation (sd )
0.14
0.14
0.20
0.20
Potential bias
(bu, bl )
Significance level
0.10
0.15
–0.008,
–0.003,
–0.008,
–0.003,
370
154
757
314
0.008
0.003
0.008
0.003
313
130
638
265
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Table 12. Probability that bias of an estimated rangewide trend
for a landbird species, surveyed by the North American Breeding Bird Survey, exceeds 0.005, 0.01, 0.02, and 0.03 when
portions of the range are not covered by the survey.
No. of
regions
useda
Proportion
of range
excluded
0.005
0.01
0.02
0.03
5
4
3
2
1
0.17
0.33
0.50
0.67
0.83
0.30
0.42
0.54
0.68
0.84
0.04
0.10
0.22
0.42
0.68
<0.01
<0.01
0.01
0.10
0.42
<0.01
<0.01
<0.01
0.01
0.22
a
Bias
Total number of U.S. Fish and Wildlife Service regions = 6.
We did not consider a significance level of 0.05
on the basis that false alarm mistakes then
become much less common than failing to detect
declines, which seems inappropriate. Even with a
significance level of 0.1 and potential bias of
±0.003, the actual probability of a false alarm
error is about 0.06 (range = 0.056–0.065 depending on actual bias), well below the nominal level.
With the level of significance set at 0.15 the actual probability of a false alarm error is about 0.09,
still much smaller than the probability of missing
a decline (0.20). This result might suggest that
the level of significance be set at 0.15; however,
setting the significance level at 0.10 is much more
common and we prefer 0.10 simply for that reason. The proposed accuracy standard is thus 80%
power to detect a 50% decline within 20 years,
acknowledging potential bias, setting the level of
significance at 0.10, and using a 2-tailed test. If
the suggested limits for potential bias with the
BBS (±0.008) are accepted, then up to 370 routes
(Table 11) with sufficient data to estimate trends
are required for species with sd ≤ 0.14 (80% of the
species in our sample), and up to 757 routes are
required for species with sd ≤ 0.20 (93% of the
species in our sample). If potential bias can be
reduced to ±0.003, then about 150 routes are
needed to achieve the accuracy target for approximately 80% of the focal species.
Proportion of the Study Area Covered.—The average estimate of the variance in true trends among
USFWS regions for numerous species was 0.0243.
We solved Expression (2) for a range of values for
G and n1 (Table 12). This analysis is approximate
for several reasons, especially because it does not
distinguish between species. Some species probably show little variation in trend across regions,
whereas in other species, the variation probably is
substantial. At present, however, little basis exists
for predicting the variation for a particular species
619
other than using methods that predict the average variation. A practical guideline for how much
of a species’ range should be covered by surveys
might be constructed as follows. Our interpretation is that bias as small as 0.005 is tolerable
(Tables 10, 11). As bias rises above 0.01, however,
a much larger sample size is needed for high confidence of achieving the accuracy target. If the
surveys cover at least two-thirds of the range
(rows 1 and 2 in Table 12), then the probability of
bias >0.01 is <0.10. If only half the range is covered, then the probability of bias >0.01 rises to
0.22. Thus, while covering all of the range clearly
is desirable, missing up to a one-third of the
range does not appear critical.
In theory, one could estimate the potential bias
due to incomplete coverage for each species and
add this component to other sources of bias.
Given the uncertainties we described, however,
we feel that a simpler rule may be more helpful.
We suggest that species for which survey coverage
is less than two-thirds of the range, or other area
of interest, be considered inadequately monitored regardless of how small the estimated standard error of the trend is. As we noted earlier,
areas not covered are large, continuous portions
of the range.
Our proposed accuracy standard is thus that
surveys cover at least two-thirds of the region of
interest and that they achieve 80% power to
detect a 50% decline within 20 years, using a 2tailed test, setting the level of significance at 0.10,
and acknowledging potential bias. Adopting this
standard means that in Expression (1), d = 20,
Zα/2 = 1.645, Zβ = 0.84, and C = –0.5. For BBS
data, as noted earlier, sd = 0.14, bl = –0.008, and bu
= 0.008 are reasonable values.
The accuracy target may need modification for
some landbird species. For example, some managers or researchers may feel that smaller
declines need to be detectable for long-lived species because these species have less ability to
recover. Care would be needed when applying
our goal to species that suffer large, natural
declines after harsh winters. Additionally, defining range for colonial species is somewhat arbitrary but could affect whether surveys covered
two-thirds of a species’ range. Another caveat is
that the standard error must be reliable. Two
colonies might have very similar trends, and the
standard error of the overall trend might thus be
very small. However, a sample of 2 colonies generally is too small to use in estimating a specieswide trend. Thus, attempts to use our monitoring
620
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
goal in applied situations may show that our goal
needs revision for some species. We suspect, however, that our goal will be appropriate for most
species and that it can be modified fairly easily
when necessary.
Question (22): Which landbird species and populations in North American warrant coverage by
monitoring programs, and which species are best
monitored by the BBS and similar programs?
The initial list of landbird species that warrant
monitoring had 425 species. Review by the committee of landbird specialists resulted in relatively
few changes. The final list had 431 species. The
breeding ranges of 264 species are largely (>67%)
within the temperate region, 39 of the species’
ranges are primarily in the northern (arctic and
boreal) regions, and 128 of the species’ ranges are
in both the temperate and northern regions. The
total number of populations was 559 (Table 13).
The committee identified 297 populations (53%
of all the landbird populations; 76% of the temperate, landbird populations) that they believed
can be best monitored with the BBS and similar
programs.
Question (33): How many of the populations best
monitored by the BBS are adequately covered
now, and how many routes would be needed to
provide adequate coverage for most of these
populations?
In Expression (1), which is based on a simple
random sample, the standard error of r (the
annual rate of change) is sd/√n. Rearranging
Expression (1) yields
.
(4)
With d = 20, C = –0.5, bu – bl = 0.016, Zα/ 2 = 1.645,
and Zβ = 0.84, as we suggested for the accuracy
target, se(r) = 0.0073. With bu = 0.003, bl = –0.003,
bu – bl = 0.006, and se(r) = 0.0113. Thus, with the
current potential bias of ±0.008, estimated trends
with standard errors ≤0.0073 achieve the accuracy
target. We noted above that achieving the accuracy target in areas one-third the size of the temperate regions of Canada and the United States
seems reasonable. A species whose range covers
the entire area needs to have a rangewide se(r)
such that with one-third of the rangewide data,
the se(r) would still be ≤0.0073. Assuming an even
distribution of routes, so that the sampling plan
Table 13. Populations of North American landbirds judged to
warrant monitoring programs.
Description
Temperate-nesting populations
Best monitored by the BBSa and
similar programs
Other populationsb
Northern-nesting populations
Total
Number
Percent
392
70
297
95
167
559
53
17
30
a
b
North American Breeding Bird Survey.
Raptors (24), nocturnal species (24), hummingbirds (14),
uplands game birds (11), and others (largely southwestern
species, 22).
may be viewed as simple random, the rangewide
standard error may be expressed as c/√nr , where c
is a constant, independent of sample size, and nr
is the rangewide sample size. The requirement
is thus that the regional se(r) = c/√ 0.33nr ≤0.0073
or that the rangewide standard error c/√nr=
(√0.33) = 0.0042. Thus, the threshold se(r) for a
species whose range covers the entire study area is
0.0042. By the same rationale, a species whose
range covers two-thirds of the BBS survey area
would need a se(r) = √0.67)*0.0073 = 0.0060. We
categorized each of the species with rangewide
standard errors <0.0073 into 1 of 3 groups
depending on whether their range covered <33
(group 1), 33–67 (group 2), or >67% (group 3) of
the BBS survey area. The threshold standard
errors for these groups were 0.0073, 0.0060, and
0.0042, respectively. We then determined how
many of these species had observed, rangewide
standard errors less than the threshold for their
group. This analysis identified 124 species that
are adequately monitored according to the proposed accuracy target. This number is 42% of the
297 species for which the BBS is considered a suitable monitoring program.
If potential bias is reduced to ±0.003, then the
target rangewide standard errors are 0.0113 for
species in group 1, 0.0092 for group 2, and 0.0065
for group 3. Using these target standard errors,
the number of adequately monitored species is
183, or 62% of the species considered well suited
to monitoring with the BBS. Thus, reducing
potential bias from ±0.008 to ±0.003 would increase the number of adequately monitored species by about 50% (from 124 to 183). In contrast,
doubling the number of BBS routes would increase the number of adequately monitored species by only about 30% (from 124 to 162). This
analysis, like the analysis described above of how
many routes are needed, shows that achieving a
J. Wildl. Manage. 68(3):2004
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
621
Table 14. Estimated number of North American Breeding Bird Survey (BBS), or similar, routes needed to achieve adequate coverage of ≥80% of the North American landbirds that warrant monitoring, if potential bias is reduced to ±0.003 and if bias remains
at its current level of ±0.008. The analysis assumes equal density of routes throughout the BBS survey area and that the goal is
achieving the accuracy target for a species’ entire range or in any area one-third the size of the temperate portions of Canada
and the United States, whichever is smaller.
Province/state
Alabama
Alberta
Arizona
Arkansas
B.C.b
California
Colorado
Connecticut
Delaware
Florida
Georgia
Idaho
Illinois
Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Manitoba
Maryland
Massachusetts
Michigan
Minnesota
Mississippi
Missouri
Montana
Nebraska
Nevada
Current no.
of routesa
83
90
65
36
81
185
107
18
11
93
65
59
82
46
35
38
43
44
61
46
64
23
81
77
35
54
62
43
27
a Indicates routes conducted
b British Columbia.
c Prince Edward Island.
Needed if bias =
±0.003
±0.008
72
189
158
73
178
219
145
7
3
78
81
116
78
51
78
114
56
64
45
95
14
11
80
117
66
97
205
107
154
172
454
379
176
427
526
347
17
7
186
195
278
188
121
188
274
134
153
107
229
32
27
193
282
159
233
491
258
369
Province/state
New Brunswick
New Hampshire
New Jersey
New Mexico
New York
Newfoundland
North Carolina
North Dakota
Nova Scotia
Ohio
Oklahoma
Ontario
Oregon
Pennsylvania
P.E.I.c
Quebec
Rhode Island
Saskatchewan
South Carolina
South Dakota
Tennessee
Texas
Utah
Vermont
Virginia
Washington
West Virginia
Wisconsin
Wyoming
Current no.
of routesa
25
23
31
62
110
14
60
45
27
76
61
105
109
108
4
73
5
45
24
50
48
170
73
24
72
89
50
72
103
Needed if bias =
±0.003
±0.008
34
13
10
169
67
40
68
98
27
57
97
148
135
63
3
110
1
120
43
107
58
367
118
13
55
93
34
78
136
82
31
25
406
162
97
163
236
64
137
233
354
324
151
7
65
3
289
103
257
140
881
283
32
133
224
81
187
326
by BBS.
general increase in the number of BBS routes is
not a very effective way to increase the number of
adequately monitored species.
With potential bias of ±0.008, the number of
routes required for adequate coverage of ≥80% of
the landbirds that warrant coverage was 370
(Table 11). With potential bias reduced to ±0.003,
the number of required routes was 150. We used
these values (n = 370 and n = 150), along with
species-specific values for Ai and yi , in Expression
(3) to estimate the density of routes needed for
the sample of species best suited to monitoring
using the BBS. The eightieth quantiles for the route
density were 12 routes/10,000 km2 with potential
bias of ±0.008 and 5 routes/10,000 km2 with potential bias of ±0.003. We used these densities to estimate the minimum number of BBS routes that
would be needed in each province and state to
achieve the monitoring objective for ≥80% of the
species best suited to monitoring with the BBS.
The results (Table 14) showed that 5 provinces
or states have the required number of routes at
present (potential bias of ±0.008); but if potential
bias can be reduced to ±0.003, then 19 provinces
and states have the required number of routes.
The total number of surveyed routes currently is
about 3,640. If potential bias can be reduced to
±0.003, the needed number is 5,106 (a 40% increase); whereas if bias is not reduced, then
>12,000 routes would be needed to meet the
accuracy target for 80% of the species that warrant monitoring. If bias were reduced to ±0.003,
then nearly enough routes are being surveyed in
most states and provinces, but the number of
BBS or similar routes needs to be more than doubled in 15 states and provinces (Table 12).
In summary, at present, only about 42% of the
species that are considered suited for monitoring
with the BBS and similar programs are adequately
monitored under the standard we propose.
622
TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
Increasing the number of routes will not by itself
greatly increase the number of adequately monitored species. However, if measures to reduce
potential bias are implemented, then a 40% increase in the number of BBS routes, concentrated in 15 states and provinces, would result in adequate coverage for approximately 80% of the
species that are suitable for monitoring with the
BBS and similar programs.
MANAGEMENT AND RESEARCH
IMPLICATIONS
Reducing Bias
The difficulty of setting limits on bias should be
an impetus to developing and using methods that
have little or no bias. The review by O’Connor et
al. (2000) identified many modifications to the
BBS that would reduce potential bias. Several
general methods have been proposed for reducing potential bias, including distance methods
(Buckland et al. 2001), double-observer methods
(Nichols et al. 2000), removal methods (Farnsworth et al. 2002), and double sampling (Bart
and Earnst 2002). Review of these approaches is
beyond the scope of this report, but we recommend that program designers give careful consideration to whether these or other methods to
reduce potential bias may yield estimates of higher accuracy than unadjusted counts.
Increased Coordination
Many state and regional programs exist or are
being planned to monitor landbirds during the
breeding season. For example, monitoring programs are scheduled for 9 of the 11 westernmost
states (excluding Alaska and Hawaii), and at least
6 large, long-term programs are coordinated by the
U.S. Forest Service (C. Hargis, U.S. Forest Service, personal communication). Although these
programs generally use different methods than
those used by the BBS, if the trend estimates they
produce are reliable, then they could be combined
with the BBS trend estimate. Thus, many states and
regions already have—or soon will have—sufficient
data to meet the accuracy target. The challenge is
to integrate all of these efforts into a single, comprehensive landbird monitoring program.
Other Surveys
More than one-third of the ranges of 167 landbird species are within the northern boreal
regions. These species cannot be adequately monitored solely with temperate breeding-season sur-
J. Wildl. Manage. 68(3):2004
veys. Breeding-season surveys in northern regions
and/or surveys at other times of year must be
implemented. A major effort is thus needed to
determine which approach should be implemented and to design the needed new surveys.
Approximately 95 temperate-nesting species
are unlikely to be covered either by increased
BBS (or similar) surveys or by surveys in northern
areas (Table 13). Extensive surveys exist for many
upland game species and raptors (mainly at
migration stations), and several regional programs exist for nocturnal and colonial species.
Much more work is needed, however, to develop
comprehensive programs for species not suited
to coverage by the BBS.
Increased Funding for Centralized
Programs
The BBS is coordinated by 2 federal employees,
1 in Canada and 1 in the United States. Improvements, such as developing ways to reduce potential bias and implementing new surveys, will require a substantial expansion of the resources that
support these centralized programs. In designing
long-term surveys, biologists can help implement
these ideas by: (1) adopting clear accuracy targets,
such as we suggested; (2) insisting that potential
bias be estimated in survey design and analysis,
and that effects of bias be acknowledged when
making inferences; and (3) encouraging preparation and peer review of survey protocols before
the surveys are implemented.
ACKNOWLEDGMENTS
We thank D. Krueper, T. Leukering, B. Peterjohn, and K. Rosenberg for reviewing and revising the list of species that warrant monitoring
and for identifying the surveys most suitable for
each species, and B. Peterjohn and 2 anonymous
reviewers for extensive comments on past drafts
of this manuscript. D. McNicol provided the
range maps from Project WILDSPACE. This project was supported by the U.S. Geological Survey
and by Bird Studies Canada.
LITERATURE CITED
ASKINS, R. A. 2000. Restoring North America’s birds.
Yale University Press, New Haven, Connecticut, USA.
BAIN, L. L., AND M. ENGELHARDT. 1987. Introduction to
probability and mathematical statistics. Doxbury
Press, Boston, Massachusetts, USA.
BART, J., B. COLLINS, AND R. I. G. MORRISON. 2003. Trend
estimation using a linear model. Condor 105:367–372.
———, AND S. L. EARNST. 2002. Double sampling to estimate bird density and population trends. Auk
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(Appendix on next page)
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APPENDIX A: DERIVATIONS AND
ADDITIONAL EXPLANATIONS
J. Wildl. Manage. 68(3):2004
where Z is a standard normal variable. By the
same rationale, for r > 1, from Expression (A.3),
Formulas for Power
We follow the notation of Steel and Torrie
(1980:114–119): α and β are the Type I and Type
II errors, respectively (power = 1– β), and Zα satisfies the expression P(Z > Zα ) = α, where Z is a
standard normal variable. Initially, we assume large
sample sizes so that Z-values may be used. Let R
equal the true rate of change in population size,
r equal the estimate of R, and se(r) be the estimated standard error of r. Let bias in r be E(r) –
R = B. Since r and R are expressed as annual rates
of change, B is a small number such as –0.01 or
0.005. We assume that bl < B < bu .
If the trend estimate is <1, we only wish to conclude that the population is declining if the observed decline is significantly less than 1 minus
the lower endpoint for B; that is, if
,
(A.1)
where Zα/2 is the critical value. We would thus
consider the result significant if and only if
.
(A.2)
By the same reasoning, if the point estimate is >1,
we would only consider it significant if
.
(A.3)
. (A.6)
Omitting bl and bu is equivalent to ignoring the
bias. In this case, the probability of a Type I error
(rejecting the null hypothesis when it is true),
since R = 1, is
. (A.7)
Power is usually defined as the probability of
rejecting the null hypothesis when it is false, and
then 1 of the outcomes, Expression (A.5) or
(A.6) in our case, is usually assumed to be sufficiently unlikely to be ignored. Thus, if R > 1 and
power is high enough to be interesting, the usual
assumption would be that a significant r < 1
would hardly ever occur, so the probability of this
event could be ignored. This is not necessarily
true when bias is present but is ignored. We
therefore defined power as the probability of
obtaining a significant r < 1 if R < 1 or a significant r > 1 if R > 1. This distinction only mattered
in a few analyses, but when it did, we referred to
the power to detect a decline (if R < 1) or an
increase (if R > 1). Thus,
The probability of rejecting the null hypothesis
that R = 1 is thus the sum of the probabilities of
the events described by Expressions (A.2) and
(A.3). First, consider
(A.8)
and
.
(A.4)
.
To calculate this probability, we manipulate the
inequality so that the left side becomes a standard normal variable. Because the value of bl does
not depend on the sample, E(r – 1 – bl ) = R – 1 +
B – bl. For the same reason, SE(r – 1 – bl ) = se(r).
Thus, for r < 1, from Expression (A.2), we obtain
,
(A.5)
(A.9)
When bl and bu are defined, then a further simplification is possible. In Expression (A.5) for r <
1, bl < B so B – bl = |B – bl |, and we may write
. (A.10)
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TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
625
In Expression (A.6) for r > 1, bu > B, so (bu – B) =
|bu – B | or |B – bu | because the order in an
absolute difference does not matter. Thus,
(A.15)
. (A.11)
Expressions (A.10) and (A.11) may be used to calculate a general expression for the probability of
making a Type I error when bias is acknowledged:
the last line holding because R < 1. If R > 1, then
from Expression (A.11), we may write
(A.16)
(A.12)
because R > 1. Thus, a general formula may written for power,
, (A.17)
The maximum occurs when B = bl or B = bu , in
which case
. (A.13)
where bp (p for potential) = bl for R < 1 and = bu
for R > 1. The maximum value of |B – bp | is bu –
bl , so this value should be used to ensure that
power will be as high as predicted by Expression
(A.17). We thus use
The minimum occurs when B = (bu + bl )/ 2, in
which case |B – bl | = |B – bu | = (bu – bl )/2, so
.
(A.14)
. (A.18)
Continuing with the derivation, if power is to be
at least 1 – β, then it must be true that
. (A.19)
Expressions (A.13) and (A.14) thus provide the
limits for the probability of a Type I error when
bias is between bl and bu and these limits are
acknowledged in the analysis.
If power is fairly high (e.g., <0.50) then the
usual assumption that 1 of the alternatives, significant r < 1 or significant r > 1, can be ignored
is reasonable. If R < 1, then from Expression
(A.10), we may write
If simple random sampling is employed to select
survey locations, and the trend estimation method
of Bart et al. (2003) is used to calculate r, then the
se(r) may be written
,
(A.20)
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TRENDS IN LANDBIRD ABUNDANCE • Bart et al.
where
E[Var(ri )]. Thus, an unbiased estimate of V(Ri ) is
,
(A.21)
bi and ymid,i are the slope and midpoint of the
regression for site i (Bart et al. 2003). The BBS
locations are a stratified sample, not a simple random sample, but we used a method described by
Cochran (1977:136) to estimate the population
variances and covariances.
Expressing R in terms of total change makes
the results above more general. Let C be the total
change during the survey’s d years. If the survey
result declined 25%, then C would be –0.25. The
relationship between C and R is C = R d – 1, and
thus R = (C + 1)1/d.
In Expression (A.19), substituting (V/n)0.5 for
se(r), (C + 1)1/d for R, and solving for n yields
,
(A.22)
where sd = v 0.5. Expression (A.22) is Expression (1)
from the body of the paper.
When t -values will be used in the analyses, Steel
and Torrie (1980:118) recommend multiplying
the estimated power, obtained assuming Z-values
are used, by (df + 3)/(df + 1), where df = degrees
of freedom. If df > 20, then the multiplier is <1.1
so it has little effect in cases of practical interest.
Derivation of Expression (2)
For a given species, let Ri = the true trend in
region i and ri = the estimate of this trend. The
variance of the ri may be expressed as
.
(A.23)
For this analysis, we defined Ri as E(ri). Making this
substitution and rearranging Expression (A22),
.
,
(A.25)
where ni = the number of regions with estimated
trends for the species. The standard deviation of
the true, region-specific trends may be estimated
as the square root of Vˆ(Ri ).
We calculated Vˆ(Ri ) using BBS data for species
recorded on >50 routes, having rangewide
SE(trend)s of <0.01, and for which trend estimates from at least 4 of the 6 U.S. Fish and Wildlife Service regions were available. These criteria
yielded 89 species. Estimates of V(Ri ) were made
using Expression (A.3) for each of the 89 species.
We found 1 clear outlier (house finch [Carpodacus mexicanus]; variance >5 times larger than the
next smaller value). Excluding this value, the
mean estimated variance was 0.024; the estimated
standard deviation was 0.016.
Suppose that a species was equally abundant in
n regions, and the standard deviation of the true,
region-specific trends was 0.016. How much bias
would be caused by estimating the rangewide trend
using data from fewer than all n regions? Let n 1 +
n 2 = n, where we use trends from n 1 regions, and
ignore the other n 2 trends in calculating the mean
trend. Let Rn = the mean of all n regional trends,
and define this as the population-wide trend (since
we assume the species to be equally abundant in
the regions). Also, let R 1 be the mean of the n 1
regional trends and R 2 be the mean of the other n 2
regional trends. The bias arising from using R 1 to
estimate Rn is R 1 – R n, so we need an expression for
this difference that allows us to calculate the probability that the bias exceeds any given amount (G)
based on the estimated variation among the
regional values, which is SD(Ri ) = 0.016. If we randomly select the n 1 regions to use in calculating R 1,
then R 1 is the mean of n 1 normal random variables. The expected value is Rn and its variance
(e.g., Bain and Engelhardt 1987:213) is σ2/n1, or
0.0243/n 2 in our case. We may therefore write
(A.24)
An unbiased estimate of V(ri ) is provided by the
sample analogue, v(ri ). An unbiased estimate of
Var(ri ) is provided by [SE(ri )]2, where SE(ri ) is obtained from BBS data. The mean of the [SE(ri )]2
of the values for regions with estimates is
,
where Z is a standard normal variable.
(A.26)